Realizing the analytic surgery group of Higson and Roe geometrically, part I: the geometric model

Abstract

We construct a geometric analog of the analytic surgery group of Higson and Roe for the assembly mapping for free actions of a group with values in a Banach algebra completion of the group algebra. We prove that the geometrically defined group, in analogy with the analytic surgery group, fits into a six term exact sequence with the assembly mapping and also discuss mappings with domain the geometric group. In particular, given two finite dimensional unitary representations of the same rank, we define a map in the spirit of η-type invariants from the geometric group (with respect to assembly for the full group C * -algebra) to the real numbers.

Relative geometric assembly and mapping cones, part I: the geometric model and applications

“…• the mapping φ * : K * (C * (Γ 1 )) → K * (C * (Γ 2 )) is (rationally) surjective, • the mapping (Bφ) * : K * (BΓ 1 ) → K * (BΓ 2 ) is (rationally) injective, • for a finite CW -pair (X, Y ), Γ 1 = π 1 (Y ), Γ 2 = π 1 (X) and φ being induced by the inclusion, the natural mapping K * (C φ ) → K * (SC * (π 1 (X/Y ))) ⊕ K * (C * (Γ 1 )) is (rationally) injective, then the following diagram commutes (rationally): (20) where the horizontal maps are the assembly maps (defined, respectively, in Sections 2 and 3) and the vertical maps are the isomorphisms defined, respectively, in Theorem 3.11 and equation (13) in Lemma 4.8.…”

“…We now turn to the proof of Theorem 5.1. The idea is to use the absolute case to prove the commutativity of the diagram (20). It remains to verify that the assumptions in Theorem 5.1 imply the commutativity of (20).…”

“…We need to show that y vanishes (rationally). Since the diagram (20) commutes in the absolute case, δ(y) = 0 ∈ K * −1 (C * (Γ 1 )). However, if φ * : K * (C * (Γ 1 )) → K * (C * (Γ 2 )) is (rationally) surjective then δ is (rationally) injective and y vanishes (rationally).…”

“…The reader can find more on this map in, for example, [20,Introduction], and the references therein. Under the isomorphisms with the analytic models, the geometrically defined assembly mapping (1) coincides with the standard construction of assembly (see [30]).…”

Positive scalar curvature and product formulas for secondary index invariants

“…Recently, this approach has also been studied by Zenobi [30] with a focus on the signature operator and secondary invariants associated to homotopy equivalences. Moreover, the product formula can be implemented using the geometric picture of the structure group due to Deeley-Goffeng [3]. Another discussion of (1.1) can be found implicitly in the work of Xie-Yu [28, pp.…”